Rats and wine bottles’ puzzle

I have tried to solve the puzzle starting with smaller numbers of wine bottles.

Let’s take 2 bottles and 1 rat.

Feed 1 bottle to the rat. This leads to 2 possible outcomes: either the rat dies or the rat doesn't die. This will tell us which of the 2 bottles is poisoned.

Depending on the rat’s life/death outcome, it would be easy to figure out which bottle was poisoned.

I have tried to show this using in table 1. Where 0 means the bottle was not fed to that rat 1 and means the bottle was fed to that rat. See table 1 below

Let’s say the rat dies that means the bottle 2 with the ‘1’ was poisoned.

Let’s say the rat survives that means the bottle 1 with the ‘0’ was not poisoned.

Let’s take 4 bottles and 2 rats.

There is one of four possibilities for the rats: See table 2 below

if neither dies, bottle 1 poisoned, if only rat 2 dies that means bottle 2 was poisoned, if only rat 1 dies that means bottle 3 was poisoned, and if both die that means bottle 4 was poisoned

Table 2 built on the same logic as table 1. Depending on which rats die and live, we can figure out which bottle was poisoned.

That means: 2² = 4.

Carrying on, with 3 rats we can actually find the poisoned bottle out of 8 bottles.

Table 3 built on the same logic as table 2. Depending on which rats die and live, we can figure out which bottle was poisoned.

That means: 2³ = 8. In this manner, we can find the poisoned bottle out of 2ⁿ bottles with n rats.

We can test 16 bottles by taking 4 rats; we can test 32 bottles by taking 5 rats for 32 bottles, and so on. 2¹⁰ = 1024 fortunately, we have 1000 bottles only.

There is only 1 rat died after testing 999 wine bottles.

Draft 1 : Unit plan Assignment

EDCP 342A Unit planning: 

Your name: Hemlatta Engineer
School, grade & course: Seaquam Secondary School, Grade 12 & Calculus.

The topic of unit: Exponential and logarithm functions

Preplanning questions:

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, and beautiful about this topic? (150 words)         The topic is included in the curriculum because Exponential and logarithm functions develop the concept of instantaneous rate of change. It is important that students learn the topic so that they can apply them to solve the growth and decay problems. I hope students will learn from the Exponent and logarithm functions unit that how to model real-world situations.         Logarithm functions are helpful when working with phenomena that have a very wide range of values that students actually work within a smaller range. The Logarithm functions that are used to measure the magnitude of earthquakes. The magnitude of an earthquake is related to how much energy-related by the quake so that we can save human lives by predicting the after sock the earthquake’s effects.          Exponential functions are helpful with phenomena that change very quickly or that grow or decay by percentage over a particular time period. In many of the sciences, certain quantities grow or decay at a rate that is proportional to their size.

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words) The Process of the project: The project is all about comparing the Exponential function and logarithm function. The project helps students to connect math to real-world situations such as the growth of population, the growth of bacteria, sea level raise, temperature raise, weather forecasting, or university fees raise.   The students will be asked: Students will pick an Exponential function or logarithm function growth situation and research real-life rates. They create an equation, write a story/ world problem, create a graph, present the PowerPoint presentation/model/poster, and compare Exponential function’s equation with a logarithm function’s equation or vice a Versa. Aims to do on their project: create the equations, compare the properties of the two functions, and make scientific conclusions based on mathematical justification. When I will give the project to students, I will allow students to work either in pairs, or small groups, or individually. I will give clear instructions on what is expected to do during the project and give a couple of days time to think about the project and ask them how they want to do the project, how they want to work. I will check the status of their process on the project regularly. I will give the due date to submit the project which would be a month later, at the end of the unit. I will assess the project when they will share their projects with the class as a group and I will check do students cover the skills/aims which are essential to complete the project.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) I will build a fair and well-rounded assessments and evaluation plan for this unit by below mentioned three assessment modes. Pre-assessment encourage students to evaluate their current thinking about the learning topic.  Students will find out what they already know about the topic and what they do not know about the topic before trying to teach them a new topic. Informal/observational assessments push students to recognize the conceptual change. I would like to introduce a new topic through student-centered problem-solving activity so that they can experience the topic. Summative Assessments after the lessons, during the mid-unit, and at the end of the unit.

Elements of my unit plan: Give a numbered list of the topics of the 10 lessons in this unit in the order I would teach them.

Lesson	Topic
1	Laws of exponents and logarithms
2	Exponential of functions
3	Derivatives of Exponential functions
4	Properties of  the Derivatives of Exponential functions
5	Logarithmic functions
6	Derivatives of logarithmic functions
7	Natural logarithm and common logarithm
8	Properties of the derivatives of logarithmic functions
9	Exponential growth & decay
10	Logarithmic differentiation

b) Write a detailed lesson plan for three of the lessons:

MATH LESSON PLAN 1

Subject: Math	Grade: 12	Date: Feb. 25, 2020	Duration: 80 Minutes
Lesson Overview (What this lesson is about)	This lesson will help students understand the concept of the Exponential of functions and its importance. It will also help the students to build a concrete foundation on Exponential functions by using/ analyzing the graph.
Class Profile	The class has 30 students including 5 ELL learners, 16 girls, and 14 boys.

Big Idea

Differential calculus develops the concept of instantaneous rate of change.

Curriculum Competencies (What the students will do)	Students are expected to: ·       Explore, analyze, and apply mathematical ideas using reason, technology, and other tools ·       Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about the number ·       Visualize to explore and illustrate mathematical concepts and relationships ·       Apply flexible and strategic approaches to solve problems
Content (What the students will know)	Determine the Exponential of functions by using the graphs and by using the laws of exponents Determine the Exponential of functions by using the laws of exponents
Language Objectives (What new language students will learn)	Define Exponential of functions, in academic and in layman terms Sketch the Exponential of functions’ graphs in small groups Explain & State the domain, the range, and the asymptote of the Exponential of functions to the class, both orally and in writing.

Materials and Equipment Needed for this Lesson

·       Calculus 12 textbook ·       Colored pens, pencils, eraser, grid papers, and ruler ·       Whiteboard with markers ·       Exponential of functions Worksheets including problem-solving activity

	Lesson Stages	Learning Activities	Time Allotted
1.	Warm-up    Get students’ attention, connect to previous  knowledge and explain why the topic is  Important to learn.	Problem solving activity: Students will complete the table, Graph the points and connect with a smooth curve, draw the graphs, and compare the graphs, based on the examples are provided by the teacher. Graph Graph f(x) = 2x  and f(x)=(1/2)x     I will ask students “What happens to f(x) as x increases?”  “What happens to f(x) as x decreases?” Based on students’ answers response, ask further questions to gather more information about the graphs. Questions for assessing understanding: “Does the graph intersect the x-axis? Explain how you know.” “What are the domain and range of f(x)?” *If students struggle to do the activity, more time will need to be allotted to do the activity before introducing the Exponential functions*	15 minutes
2.	Presentation Teach the new content and language.	Introduction: “What is an Exponential Function?" Graphing Exponential Functions: I will use the same activity graphs and We will take a look at the similarities and differences between the two graphs so that students can recognize the types of Exponential function. Graphing the exponential function by solving more examples.	20 minutes
3.	Practice and Production Practice, reinforcement, and extension of the new content and language.	Students work on the given worksheet in groups. ELL students will be paired with non ELL students during classwork as groups in order to ensure that they are able to work on their understanding of the exponential functions. I will walk around and give hints to groups, help them to correct any errors made. For a mistake made by most groups, gather all students and explain the concept.	30 minutes
4.	Closure	As a class, review the topic by recognizing the types of Exponential of function and solving examples on the whiteboard. I will ask students to predict if the resulting graph would be exponential growth or exponential decay, how can we correctly answer the question without actually drawing the graph?	15 minutes

Assessment/Evaluation of Students’ Learning

I will give the worksheets to the entire class so as to ensure all students work on it. Students will share their perspectives with the class to review the concepts learned that day, so their level of understanding can be increased. I will do informal assessment to make sure that students have a solid understanding of an exponential function in relation to a scenario as this will aid in their theoretical development of its graph and reflect on what students do not understand generally, and then review it again in the next class. I will also be able to see which students are not good at the lesson, so I can assist them further.

MATH LESSON PLAN 2: Derivatives of Exponential functions  

Subject: Math	Grade: 12	Date: Feb. 26, 2020	Duration: 80 Minutes
Lesson Overview (What this lesson is about)	This lesson will help students understand the concept of the Derivatives of Exponential of functions and its importance. It will also help the students to build a concrete foundation on the Derivatives of Exponential of functions by using/analyzing the graph.
Class Profile	The class has 30 students including 5 ELL learners, 16 girls, and 14 boys.

Big Idea

Differential calculus develops the concept of instantaneous rate of change.

Curriculum Competencies (What the students will do)	Students are expected to: ·       Explore, analyze, and apply mathematical ideas using reason, technology, and other tools ·       Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about the number ·       Visualize to explore and illustrate mathematical concepts and relationships ·       Apply flexible and strategic approaches to solve problems
Content (What the students will know)	Determine the Derivatives of Exponential of functions by using the graphs   Determine the Derivatives of Exponential of functions by using the lows of exponents
Language Objectives (What new language students will learn)	Define  the Derivatives of Exponential of functions, in academic and in layman terms Apply the product and chain rules for the derivatives of the exponential of functions to the class, both orally and in writing.

Materials and Equipment Needed for this Lesson

·       Calculus 12 textbook ·       Colored pens, pencils, eraser, grid papers, and ruler ·       Whiteboard with markers ·       Derivatives of Exponential of functions Worksheets including discussion activity

	Lesson Stages	Learning Activities	Time Allotted
1.	Warm-up    Get students’ attention, connect to previous  knowledge and explain why the topic is  Important to learn.	Discussion activity: Students will discuss how to apply the limit definition of Derivatives of Exponential of functions based on the example is provided by the teacher.  Graph f(x) = 2x   I will ask students “What happens to f(x + h) as we put x = 2x  ?”  “What happens to f(x) as we put x= b x?” Based on students’ answers response, ask further questions to gather more information about the definition. Questions for assessing understanding: “Does the factor in front of the limit depend on the limit? Explain how you know.” “What is the difference between the product rule and the chain rule?” *If students struggle to do the activity, more time will need to be allotted to do the activity before introducing the Exponential functions*	15 minutes
2.	Presentation Teach the new content and language.	Introduction: “What is Derivative of an Exponential Function?" I will use the product rule and the chain rule to determine the derivatives of Exponential of function by solving the examples.	20 minutes
3.	Practice and Production Practice, reinforcement, and extension of the new content and language.	Students work on the given worksheet in groups. ELL students will be paired with non ELL students during classwork as groups in order to ensure that they are able to work on their understanding of the derivatives of exponential functions. I will walk around and give hints to groups, help them to correct any errors made. For a mistake made by most groups, gather all students and explain the concept.	30 minutes
4.	Closure	As a class, review the topic by class discussion Activity to how to determine derivatives of Exponential function and solving examples on the whiteboard. I will assign homework to solve examples for practice. I will let them know why it is important to understand the concept of Derivatives of the functions.	15 minutes

Assessment/Evaluation of Students’ Learning

I will give the discussion activity to the entire class so as to ensure all students work on it. Students will share their perspectives with the class to review on the concepts learned that day, so their level of understanding can be increased. I will do an informal assessment and reflect on what students do not understand generally, and then review it again in the next class. I will also be able to see which students are not good at the lesson, so I can assist them further.

MATH LESSON PLAN 3

Subject: Math	Grade: 12	Date: Feb. 27, 2020	Duration: 80 Minutes
Lesson Overview (What this lesson is about)	This lesson will help students understand the concept of the Properties of the Derivatives of Exponential of functions and its importance. It will also help the students to build a concrete foundation on Exponential of functions by applying its properties to the real world situations.
Class Profile	The class has 30 students including 5 ELL learners, 16 girls, and 14 boys.

Big Idea

Differential calculus develops the concept of instantaneous rate of change.

Curriculum Competencies (What the students will do)	Students are expected to: ·       Explore, analyze, and apply mathematical ideas using reason, technology, and other tools ·       Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about the number ·       Visualize to explore and illustrate mathematical concepts and relationships ·       Apply flexible and strategic approaches to solve problems
Content (What the students will know)	Determine the 8 Properties of the Derivatives of Exponential of functions by sketching the graphs
Language Objectives (What new language students will learn)	Define the 8 Properties of the Derivatives of Exponential of functions, in academic and in layman terms by sketching the Exponential of functions’ graphs Explain & State the domain, the range, the asymptote, the interval of increase or decrease, the extreme value, concavity, and the sketch of the curve of the Exponential of functions to the class, both orally and in writing.

Materials and Equipment Needed for this Lesson

·       Calculus 12 textbook ·       Colored pens, pencils, eraser, grid papers, and ruler ·       Whiteboard with markers ·       Properties of the derivatives of the Exponential of functions Worksheets including problem-solving activity

	Lesson Stages	Learning Activities	Time Allotted
1.	Warm-up    Get students’ attention, connect to previous  knowledge and explain why the topic is  Important to learn.	Problem solving activity: Students will complete the table, Graph the points and connect with a smooth curve, draw the graphs, and compare the graphs, based on the examples are provided by the teacher. Graph f(x)=2x/2 and f(x) = (1/2)-2x I will ask students “Does the curve symmetry to any axis?” “Does the graph of f(x) have  Y-intercept?” Based on students’ answers response, ask further questions to gather more information about the graphs. Questions for assessing understanding: “Does the graph intersect the x-axis? Explain how you know.” “What is the domain of f(x)?” *If students struggle to do the activity, more time will need to be allotted to do the activity before introducing the Exponential functions*	15 minutes
2.	Presentation Teach the new content and language.	Introduction: The 8 Properties of the Derivatives of an Exponential Function Graphing Exponential Functions: I will use the same graphs that students sketch during the activity. We will state the domain, the range, the asymptote, the interval of increase or decrease, the extreme value, concavity, and the sketch of the curve as the properties of the derivatives of Exponential function.	20 minutes
3.	Practice and Production Practice, reinforcement, and extension of the new content and language.	Students work on the given worksheet in groups. ELL students will be paired with non ELL students during classwork as groups in order to ensure that they are able to work on their understanding of the exponential functions. I will walk around and give hints to groups, help them to correct any errors made. For a mistake made by most groups, gather all students and explain the concept.	30 minutes
4.	Closure	As a class, review the topic by asking the inquiry questions during the discussion Activity to how to state the properties of the derivative of Exponential of function and solving examples together, on the whiteboard.	15 minutes

Assessment/Evaluation of Students’ Learning

I will give the problem-solving activity and discussion activity to the entire class so as to ensure all students work on it. Students will share their perspectives with the class to review on the concepts learned that day, so their level of understanding can be increased. I will do a summative assessment and reflect on what students do not understand generally, and then review it again on the next class. I will also be able to see which students are not good at the lesson, so I can assist them further.